I'm raising my kids in such a way that I totally trust their judgment about what they choose to do - whether that's staying up 'til 3, or what websites they visit. Do they make mistakes? Yup. It's part of learning. BUT, we're close enough that I can sense when something's off with them. I've been respectful enough of who they are, and their choices, that they come to me to talk about it, or, they're responsive and open when I come to them.
I am their friend, absolutely, AND their parent. I don't need to control them, because I trust their process of learning self-control. Does my son visit sites that would make me blush? Undoubtedly. But I'm not here to control what he uses for a masturbatory aid, that's a deeply personal decision. Is there sick stuff out there? Yes, and when my son clicks over to it, he clicks away, because he doesn't like how it makes him feel.
He has told me when people he's gaming with have made inappropriate comments, and I've helped him work through that. He's come to me when things started feeling "oogly" when he was chatting with an online friend. That "friend" was probably a predator, but because I've allowed my son to be in touch with his gut feelings on things since he was tiny, he stopped things when it did not feel right.
If you're connected and close with your kids (i.e., their friend), they aren't alone when they venture out into the online world.
The mistrust shown by monitoring sites, looking at histories, limiting times, etc. drives a wedge between kids & parents. That's where the problem comes in, not from the internet.